Since \(\mathcal {M}_h\)  is a finite-dimensional linear space,
any grid function \(v\in \mathcal {M}_h\)  and any linear operator \(Q:\mathcal {M}_h\rightarrow \mathcal {M}_h\) 
can be regarded as a vector in \(\mathbb {R}^{M^2}\)  and a matrix in \(\mathbb {R}^{M^2\times M^2}\) , respectively.
We still use the notations \(\Vert \cdot \Vert \)  and \(\Vert \cdot \Vert _\infty \) 
to denote the matrix induced-norms consistent with \(\Vert \cdot \Vert \)  and \(\Vert \cdot \Vert _\infty \)  defined for vectors before, respectively.
By regarding \(\Delta _h\)  as a linear operator,
we know that \(\Delta _h\)  is the generator of a contraction semigroup on \(\mathcal {M}_h\)  [1]}.
Instead, by viewing \(\Delta _h\)  as a matrix,
it is weakly diagonally dominant with all diagonal entries negative.
Moreover, we have the following useful estimate
and the proof can be found in [2]}.
